v2007.11.26 - Convex Optimization

7.2. SECOND PREVALENT PROBLEM: 473Our solution to this second problem prevalent in the literature requiresmeasurement matrix H to be nonnegative;H = [h ij ] ∈ R N×N+ (1183)If the H matrix given has negative entries, then the technique of solutionpresented here becomes invalid. As explained in7.0.1, projection of Hon K = S N h ∩ R N×N+ (1133) prior to application of this proposed solution isincorrect.7.2.1 Convex caseWhen ρ = N − 1, the rank constraint vanishes and a convex problememerges: 7.12minimize◦√Dsubject to‖ ◦√ D − H‖ 2 F◦√D ∈√EDMN⇔minimizeD∑ √d ij − 2h ij dij + h 2 iji,jsubject to D ∈ EDM N (1184)For any fixed i and j , the argument of summation is a convex functionof d ij because (for nonnegative constant h ij ) the negative square root isconvex in nonnegative d ij and because d ij + h 2 ij is affine (convex). Becausethe sum of any number of convex functions in D remains convex [46,3.2.1]and because the feasible set is convex in D , we have a convex optimizationproblem:minimize 1 T (D − 2H ◦ ◦√ D )1 + ‖H‖ 2 FD(1185)subject to D ∈ EDM NThe objective function being a sum of strictly convex functions is,moreover, strictly convex in D on the nonnegative orthant. Existenceof a unique solution D ⋆ for this second prevalent problem depends uponnonnegativity of H and a convex feasible set (3.1.2). 7.137.12 still thought to be a nonconvex problem as late as 1997 [269] even though discoveredconvex by de Leeuw in 1993. [68] [39,13.6] Yet using methods from3, it can be easilyascertained: ‖ ◦√ D − H‖ F is not convex in D .7.13 The transformed problem in variable D no longer describes √ Euclidean projection onan EDM cone. Otherwise we might erroneously conclude EDM N were a convex bodyby the Bunt-Motzkin theorem (E.9.0.0.1).